Page 261 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 261
COMBINATORIAL DESIGNS (MS-16)
fractal image patterns. It constitutes a new Latin square isomorphism invariant which is ana-
lyzed in this talk for characterizing isomorphism classes of non-idempotent Latin squares in an
efficient computational way.
References
[1] V. Dimitrova, S. Markovski, Classification of quasigroups by image patterns. In: Proceed-
ings of the Fifth International Conference for Informatics and Information Technology,
Bitola, Macedonia, 2007; 152–160.
[2] R. M. Falcón, Recognition and analysis of image patterns based on Latin squares by
means of Computational Algebraic Geometry, Mathematics 9 (2021), paper 666, 26 pp.
[3] R. M. Falcón, V. Álvarez, F. Gudiel, A Computational Algebraic Geometry approach
to analyze pseudo-random sequences based on Latin squares, Adv. Comput. Math. 45
(2019), 1769–1792.
Novák’s conjecture on cyclic Steiner triple systems and its generalization
Tao Feng, tfeng@bjtu.edu.cn
Beijing Jiaotong University, China
Coauthors: Daniel Horsley, Xiaomiao Wang
Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v ≡ 1
(mod 6), it is always possible to choose one block from each block orbit so that the chosen
blocks are pairwise disjoint.
In this talk, we shall consider the generalization of this conjecture to cyclic (v, k, λ)-designs
with 1 ≤ λ ≤ k − 1. Superimposing multiple copies of a cyclic symmetric design shows that
the generalization cannot hold for all v, but we conjecture that it holds whenever v is sufficiently
large compared to k. We confirm that the generalization of the conjecture holds when v is prime
and λ = 1 and also when λ ≤ (k−1)/2 and v is sufficiently large compared to k. As a corollary,
we show that for any k ≥ 3, with the possible exception of finitely many composite orders v,
every cyclic (v, k, 1)-design without short orbits is generated by a (v, k, 1)-disjoint difference
family.
An Evans-style result for block designs
Daniel Horsley, danhorsley@gmail.com
Monash University, Australia
Coauthor: Ajani De Vas Gunasekara
A now-proven conjecture of Evans states that any partial latin square with at most n − 1 filled
cells can be completed to a latin square. This is sharp: there are uncompletable partial latin
squares with n filled cells. This talk will discuss the analogous problem for block designs.
An (n, k, 1)-design is a collection of k-subsets (blocks) of a set of n points such that each
pair of points occur together in exactly one block. If this restriction is relaxed to require only
that each pair of points occur together in at most one block we instead have a partial (n, k, 1)-
design. I will outline a proof that any partial (n, k, 1)-design with at most n−1 − k + 1 blocks
k−1
259
fractal image patterns. It constitutes a new Latin square isomorphism invariant which is ana-
lyzed in this talk for characterizing isomorphism classes of non-idempotent Latin squares in an
efficient computational way.
References
[1] V. Dimitrova, S. Markovski, Classification of quasigroups by image patterns. In: Proceed-
ings of the Fifth International Conference for Informatics and Information Technology,
Bitola, Macedonia, 2007; 152–160.
[2] R. M. Falcón, Recognition and analysis of image patterns based on Latin squares by
means of Computational Algebraic Geometry, Mathematics 9 (2021), paper 666, 26 pp.
[3] R. M. Falcón, V. Álvarez, F. Gudiel, A Computational Algebraic Geometry approach
to analyze pseudo-random sequences based on Latin squares, Adv. Comput. Math. 45
(2019), 1769–1792.
Novák’s conjecture on cyclic Steiner triple systems and its generalization
Tao Feng, tfeng@bjtu.edu.cn
Beijing Jiaotong University, China
Coauthors: Daniel Horsley, Xiaomiao Wang
Novák conjectured in 1974 that for any cyclic Steiner triple systems of order v with v ≡ 1
(mod 6), it is always possible to choose one block from each block orbit so that the chosen
blocks are pairwise disjoint.
In this talk, we shall consider the generalization of this conjecture to cyclic (v, k, λ)-designs
with 1 ≤ λ ≤ k − 1. Superimposing multiple copies of a cyclic symmetric design shows that
the generalization cannot hold for all v, but we conjecture that it holds whenever v is sufficiently
large compared to k. We confirm that the generalization of the conjecture holds when v is prime
and λ = 1 and also when λ ≤ (k−1)/2 and v is sufficiently large compared to k. As a corollary,
we show that for any k ≥ 3, with the possible exception of finitely many composite orders v,
every cyclic (v, k, 1)-design without short orbits is generated by a (v, k, 1)-disjoint difference
family.
An Evans-style result for block designs
Daniel Horsley, danhorsley@gmail.com
Monash University, Australia
Coauthor: Ajani De Vas Gunasekara
A now-proven conjecture of Evans states that any partial latin square with at most n − 1 filled
cells can be completed to a latin square. This is sharp: there are uncompletable partial latin
squares with n filled cells. This talk will discuss the analogous problem for block designs.
An (n, k, 1)-design is a collection of k-subsets (blocks) of a set of n points such that each
pair of points occur together in exactly one block. If this restriction is relaxed to require only
that each pair of points occur together in at most one block we instead have a partial (n, k, 1)-
design. I will outline a proof that any partial (n, k, 1)-design with at most n−1 − k + 1 blocks
k−1
259
